Jorgensen's Chapter 4: Graph Theory for the Testers

Chapter 4

GraphGraph Theorytheory forfor TestersTesters

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Linear Graph Theory

• A branch of topology—focus on connections • (undirected) Graphs – nodes, edges, matrices – degree of a node – Paths – Components • Directed graphs – nodes, edges – indegree, outdegree – paths and semi-paths – n-connectedness – cyclomatic number – strong and weak components – Program Graphs

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Biblical advice…

“Test everything; keep that which is good”

1 Thessalonians 5:21

[Graph theory will help us understand the “everything” part.]

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Linear Graphs

Definition 1: A graph G = (V, E) is composed of a finite (and nonempty) set V of nodes and a set E of unordered pairs of nodes.

• When drawn, graphs usually show nodes as circles, and edges as lines. • Leonhard Euler developed graphs as a way to solve the “Bridges of Königsberg” puzzle in 1735. • (Königsberg, Prussia is now Kaliningrad, Russia)

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Bridges of Königsberg Puzzle (http://en.wikipedia.org/wiki/Seven_Bridges_of_Köninigsberg)

• Take a walk around the city, traversing each bridge exactly once. • Euler’s formulation made the problem easier to solve. • (It is impossible) • Nice example of using math to solve an abstract problem.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Linear Graph (running example)

n1 e1 n2 e4 n5

n3 e3 n4 n7

V = {n1, n2, n3, n4, n5, n6, n7} E = {e1, e2, e3, e4, e5} = {(n1, n2), (n1, n4), (n3, n4), (n2, n5), (n4, n6)}

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Degree of a Node

Definition 2: The degree of a node in a graph is the number of edges that have that node as an endpoint.

• We write deg(n) for the degree of node n. • “popularity” of a node in a graph • Social scientists use graphs to describe – social interactions – friendship – communication

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Degree of a Node

n1 e1 n2 e4 n5 deg(n1) = 2 deg(n2) = 2 deg(n3) = 1 e2 deg(n4) = 3 deg(n5) = 1 deg(n6) = 1 n3 e3 n4 n7 deg(n7) = 0

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Incidence Matrix of a Graph

Definition 3: The incidence matrix of a graph G = (V, E) with m nodes and n edges is an m by n matrix, where the element in row i, column j is a 1 if and only if node i is an endpoint of edge j; otherwise, the element is 0.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Sample Incidence Matrix (What must be true about columns in an incidence matrix?)

n1 n2 n5 e1 e2 e3 e4 e5 e1 e4 n1 1 1 0 0 0 n2 1 0 0 1 0 e2 n3 0 0 1 0 0 n3 e3 n4 n7 n4 0 1 1 0 1 n5 0 0 0 1 0 e5 n6 0 0 0 0 1 n6 n7 0 0 0 0 0

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Adjacency Matrix of a Graph

Definition 4: The adjacency matrix of a graph G = (V, E) with m nodes is an m by m matrix, where the element in row i, column j is a 1 if and only if an edge exists between node i and node j; otherwise, the element is 0.

• This matrix can be used to answer questions about a program... – reachability – points affected by a fault – connectedness in a program • Question: Why is this adjacency matrix symmetric?

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Sample Adjacency Matrix

n1 n2 n3 n4 n5 n6 n7 n1 e1 n2 e4 n5 n1 0 1 0 1 0 0 0

n2 1 0 0 0 1 0 0 e2 n3 0 0 0 1 0 0 0 n3 e3 n4 n7 n4 1 0 1 0 0 1 0

n5 0 1 0 0 0 0 0 e5

n6 0 0 0 1 0 0 0 n6

n7 0 0 0 0 0 0 0

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Paths in a Graph

Definition 5: A path is a sequence of edges such that, for any adjacent pair of edges ei, ej in the sequence, the edges share a common (node) endpoint.

• Paths capture/express connectivity. • Paths can be described by – sequences of nodes, or – sequences of edges

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Some Paths

path between – n1 and n5 n1, n2, n5 e1, e4 – n6 and n5 n6, n4, n1, n2, n5 e5, e2, e1, e4 – n3 and n2 n3, n4, n1, n2 e3, e2, e1

n1 e1 n2 e4 n5

n3 e3 n4 n7

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Connectedness

Definition 5: Two nodes are connected if and only if they are in the same path.

• Question: Is connectedness in a graph an equivalence relation? • From Chapter 3, to be an equivalence relation, connectedness must be – reflexive – symmetric – transitive

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Components of a Graph

Definition 6: A component of a graph is a maximal set of connected nodes.

• Question: Does connectedness help define components? • Components are “maximal”

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Components of “our” Graph

n1 e1 n2 e4 n5

n3 e3 n4 n7

C1 = {n1, n2, n3, n4, n5, n6} C2 = {n7}

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Condensation Graphs

Definition 7: Given The condensation graph of a graph G = (V, E) is formed by replacing each component of G by a condensing node.

Questions: – Draw the condensation graph of our running example. – Can there be any edges in a condensation graph?

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Cyclomatic Number of a Graph

Definition 8: The cyclomatic number of a graph G is given by V(G) = e – n + p, where – e is the number of edges in G – n is the number of nodes in G – p is the number of components in G

• Cyclomatic complexity pertains to both ordinary and directed graphs (next topic). • V(G) is sometimes called McCabe Complexity after Thomas McCabe. • (not very useful on our running example)

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Cyclomatic Complexity of “our” Graph

n1 e1 n2 e4 n5 V(G) = e – n + p e2 = 5 – 7 + 2 n3 e3 n4 n7 = 0

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graphs

Definition 9: A directed graph (or digraph) D = (V, E) consists of a finite set V of nodes, and a set E of edges, where each edge ek = is an ordered pair of nodes ni, nj ϵ V.

• is an ordered pair • (x, y) is an unordered pair • For an edge in a directed graph, node a is the initial node, and node be is the terminal node..

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (revised example)

n1 e1 n2 e4 n5

n3 e3 n4 n7

V = {n1, n2, n3, n4, n5, n6, n7} E = {e1, e2, e3, e4, e5} = {, , , , }

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Indegrees and Outdegrees of a Node

• Definition 10: The indegree of a node in a directed graph is the number of distinct edges that have the node as a terminal node. • Definition 11: The outdegree of a node in a directed graph is the number of distinct edges that have the node as a start point.

• We write indeg(n) and outdeg(n).

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Indegrees and Outdegrees

n1 e1 n2 e4 n5 indeg(n1) = 0 outdeg(n1) = 2 indeg(n2) = 1 outdeg(n2) = 1 e2 indeg(n3) = 0 outdeg(n3) = 1 indeg(n4) = 2 outdeg(n4) = 1 n3 e3 n4 n7 indeg(n5) = 1 outdeg(n5) = 0 indeg(n6) = 1 outdeg(n6) = 0 indeg(n7) = 0 outdeg(n7) = 0 e5 n6

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Types of Nodes

Definition 12: A node with indegree = 0 is a source node. A node with outdegree = 0 is a sink node. A node with indegree ≠ 0 and outdegree ≠ 0 is a transfer node

• Question: What are the source, sink, and transfer nodes in our continuing example?

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Adjacency Matrix

Definition 13: The adjacency matrix of a directed graph D = (V, E) with m nodes is an m by m matrix A = (a(i, j)) where a(i, j) is a 1 if and only if there is an edge from node i to node j; otherwise, the element is 0

• Only in very special graphs are the adjacency matrices symmetric.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Adjacency Matrix

n1 n2 n3 n4 n5 n6 n7 n1 0 1 0 1 0 0 0 n1 e1 n2 e4 n5

n2 0 0 0 0 1 0 0 e2 n3 0 0 0 1 0 0 0 n3 e3 n4 n7

n4 0 0 0 0 0 1 0 e5

n5 0 0 0 0 0 0 0 n6

n6 0 0 0 0 0 0 0

n7 0 0 0 0 0 0 0

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Adjacency Matrix Questions

1. Can indegrees and outdegrees of a node be derived (observed) in an adjacency matrix of a directed graph? 2. Can types of nodes (source, sink, transfer) be derived (observed) in an adjacency matrix of a directed graph? 3. Can paths be derived (observed) in an adjacency matrix of a directed graph? Hint: think about powers of an adjacency matrix.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Paths in a Directed Graph

Definition 14: A (directed) path is a sequence of edges such that, for any adjacent pair of edges ei, ej, in the sequence, the terminal node of the first edge is the initial node of the second edge.

Some related definitions... – A cycle is a path in which some node is both the initial and the final node in the path. – A chain is a sequence of nodes in which every interior node has indegree = outdegree = 1 – A semipath is a sequence of edges such that at least one node is either the initial node or the terminal node of two adjacent edges in the sequence.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (new example)

n1 e1 n2 e4 n5

n3 e3 n4 n7

There is a path from n1 to n6, and there are semipaths between nodes n1 and n3, n2 and n4, and between nodes n5 and n6.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Questions

1. List the chains in our continuing n1 e1 n2 e4 n5 example. 2. What is/are the longest path(s)? 3. Are there any cycles? e2 4. How would you describe the n3 e3 n4 n7 difference between semipaths and ? e5 5. Is a n6 semipath?

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Reachability Matrix

Definition 15: The reachability matrix of a directed graph D = (V, E) with m nodes is an m by m matrix R = (r(i, j)), where r(i, j) is a 1 if and only if there is a path from node i to node j, otherwise the element is 0.

• Reachability is closely related to paths. • The reachability matrix R can be computed using the adjacency matrix A of the directed graph: – R = I + A + A2 + A3 + ... + Ak – where k is the length of the longest path in D, – I is the identity matrix, and – powers of A are computed by slightly changed matrix multiplication in which 1 + 1 = 1

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Reachability Matrix

n1 n2 n3 n4 n5 n6 n7 n1 1 1 0 1 1 1 0 n1 e1 n2 e4 n5

n2 0 1 0 0 1 0 0 e2 n3 0 0 1 1 0 1 0 n3 e3 n4 n7

n4 0 0 0 1 0 1 0 e5

n5 0 0 0 0 1 0 0 n6

n6 0 0 0 0 0 1 0

n7 0 0 0 0 0 0 1

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Compute A2

n1 n2 n3 n4 n5 n6 n7 n1 n2 n3 n4 n5 n6 n7 n1 0 1 0 1 0 0 0 n1 0 1 0 1 0 0 0 n2 0 0 0 0 1 0 0 n2 0 0 0 0 1 0 0 n3 0 0 0 1 0 0 0 n3 0 0 0 1 0 0 0 n4 0 0 0 0 0 1 0 n4 0 0 0 0 0 1 0 n5 0 0 0 0 0 0 0 n5 0 0 0 0 0 0 0 n6 0 0 0 0 0 0 0 n6 0 0 0 0 0 0 0 n7 0 0 0 0 0 0 0 n7 0 0 0 0 0 0 0

n1 n2 n3 n4 n5 n6 n7 n1

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers n-Connectedness

Definition 16: Nodes nj and nk in a directed graph are

– 0-connected iff no path (or semipath) exists between nj and nk

– 1-connected iff a semi-path but no path exists between nj and nk

– 2-connected iff a path exists from nj and nk

– 3-connected iff a path goes from nj to nk and a path goes from n to nk

• No other degrees of n-connectedness exist • A new edge, e6, is added to our continuing example. • n-connectedness has very useful expressive power.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (third version)

n1 e1 n2 e4 n5

n3 e3 n4 n7

e6 e5

Is 3-connectedness an equivalence relation?

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (third version)

n1 e1 n2 e4 n5

n3 e3 n4 n7

e6 e5

1. Find examples of a chain, a cycle, a set of 3-connected nodes

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Strong Components of a Directed Graph

Definition 17: A strong component of a directed graph is a maximal set of 3-connected nodes.

Strong components... – identify loops and isolated nodes. – lead to another form of condensation graph – support an excellent voew of testing programs with loops

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (third version)

n1 e1 n2 e4 n5

n3 e3 n4 n7

e6 e5

Strong components: S1 = {n3, n4, n6}, S2 = {n7}

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Condensation Graph of a Directed Graph

Definition 18: Given a directed graph D = (V, E), its condensation graph is formed by replacing strongly connected nodes by their corresponding strong components.

A condensation graph of a directed graph... – contains no loops, and is therefore – an Directed Acyclic Graph (DAG) – support an excellent view of testing programs with loops

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Condensation Graph of Directed Graph (third version)

n1 e1 n2 e4 n5 n1 e1 n2 e4 n5

e2 e2

n3 n4 n7 e3 S1 S2 e6 e5

Strong components: S1 = {n3, n4, n6}, S2 = {n7}

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Directed Graph (new example)

n1 e1 n2 e4 n5

n3 e3 n4 n7

V = {n1, n2, n3, n4, n5, n6, n7} E = {e1, e2, e3, e4, e5} = {(n1, n2), (n1, n4), (n3, n4), (n2, n5), (n4, n6)}

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Program Graphs

Definition 19: The program graph of a program written in an imperative programming language is a directed graph in which nodes are either entire statements or statement fragments. There is an edge from node i to node j iff node j can be executed immediately after node i).

• (The original definitions referred to nodes as entire statements, but this doesn’t fit well with modern programming languages.) • We shall use “statement fragment” to refer either to full statements or to statement fragments.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Program Graphs • When drawing a program graph, it is usually simpler to number the statement fragments. • This Figure 8.1

If-Then-Else 1 Pre-test Loop 1

1 If 2 4 1 While 2 Then 2 2 3 3 End While 4 Else 3 5 4 5 3 6 End If 6 7 4 7

Case/Switch 1 Post-test Loop 1 1 Case n Of 3 2 n=1: 2 4 6 1 Do 2 3 2 4 n=2: 3 Until 3 5 7 4 5 3 6 n=3: 7 8 8 End Case 4

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Four Graph-Based Models

• Finite State Machines • Petri Nets • Event-Driven Petri Nets • StateCharts

• These are all executable models, i.e., it is possible to build a program (an engine) to execute the model.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Finite State Machines (FSMs)

• Finite State Machines are directed graphs in which nodes are states, and edges are transitions from one state to a successor state. • Transitions are caused by – events – date conditions – passage of time (an event) • Constraints – states are mutually exclusive – only one transition can occur at a time • FSMs are ideally suited for menu-driven applications

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Garage Door Controller FSM

Input events Output events (actions) e1: depress controller button a1: start drive motor down e2: end of down track hit a2: start drive motor up e3: end of up track hit a3: stop drive motor e4: obstacle hit a4. door stops part way e5: laser beam crossed a5. door continues opening a6. door continues closing

s1: Door Up e1/a1

e3/a3

s5: s4: Door Door Stopped Going Up Closing e4/a3, a2

e1/a1 e1/a3 e1/a2 e1/a3

s3: e5/a3, a2 s6: Door Stopped Door Going down Opening

e2/a3 e1/a2

s2: Door down

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Petri Nets

• Petri Nets are bipartite directed graphs (P, T, In, Out), where – P is a set of places – T is a set of transitions – In is a mapping of places to transitions, understood as input places – Out is a mapping of to transitions to places , understood as output places

• Finite State Machines are a special case of Petri Nets.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Petri Nets (continued)

• Petri Net execution is governed by – place markings – transition enabling and firing – several strategies for transition firing sequences • Constraints – only one transition at a time can be fired – no simultaneous events • Petri Nets can express a variety of conmplex situations.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Petri Net Example

p1 p5 p2

t1 t2

p3 p4

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Petri Net Exercise

Identify the following for the sample Petri Net – the set P of places – the set T of transitions – the mapping In of inputs to transitions – the mapping out of outputs of transitions

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Event-Driven Petri Nets (EDPNs)

• Definition 20: An Event-Driven Petri Net is a tripartite- directed graph (P, D, S, In, Out) composed of three sets of nodes, P, D, and S, and two mappings, In and Out, where: – P is a set of port events – D is a set of data places – S is a set of transitions – In is a set of ordered pairs from (P ∪ D) ˣ S – Out is a set of ordered pairs from S ˣ (P ∪ D)

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Event-Driven Petri Nets (continued)

• Port events can be – input events – output events – drawn as triangles • Places are as in ordinary Petri Nets • Transition firing is a simple extension of ordinary Petri Net transition firing • (there are no output events in the example on the next slide)

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers EDPN Example

p3 d5 p4

s7 s10

p3 d6 p4

s8 s9

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers StateCharts • Created by David Harel • Harel’s goal was to combine the visual expressive power of – directed graphs – Venn diagrams • “States” can contain... – lower level states – concurrent regions • Transitions are very elaborate (see text) • StateCharts are an elegant resolution to the “Finite State Machine Explosion”

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Sample StateChart

A E

Blob (state) A contains two lower level states. Blob B contains two concurrent regions, E and F.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers Wrap-Up

• Graph theory is a powerful way to express many of the concepts of software testing. • It will be used extensively inh sequel chapters. • The graph-based models all lead to (and support) model-based testing. • Executable models support nearly automatic generation (derivation) os system level test cases.

Software Testing: A Craftsman’s Approach, 4th Edition Chapter 4 Graph Theory for Testers